Abstract: Let $D$ be a Noetherian domain and $Int(D)$ be the corresponding ring of integer-valued polynomials. We consider the prime ideals of $Int(D)$ above an height one maximal ideal $M$ of $D$ with finite residue field. Without loss of generality we may assume $D$ to be local with maximal ideal $M.$ In case $D$ is unibranch, it is known by a topological argument that these prime ideals (all maximal) are not finitely generated, using the fact there are infinitely many such primes. In case $D$ is not unibranch, there are only finitely many such primes, but we prove however here, by a similar topological argument, that these primes are again not finitely generated. It follows that $D$ is not almost strong Skolem in this case (whereas it is known to be so, in case $D$ is analytically irreducible).